Repository logo
 

An axiomatic characterization of the Brownian map

Accepted version
Peer-reviewed

Type

Article

Change log

Authors

Miller, J 
Sheffield, S 

Abstract

The Brownian map is a random sphere-homeomorphic metric measure space obtained by gluing together'' the continuum trees described by the $x$ and $y$ coordinates of the Brownian snake. We present an alternative breadth-first'' construction of the Brownian map, which produces a surface from a certain decorated branching process. It is closely related to the peeling process, the hull process, and the Brownian cactus.

Using these ideas, we prove that the Brownian map is the only random sphere-homeomorphic metric measure space with certain properties: namely, scale invariance and the conditional independence of the inside and outside of certain ``slices'' bounded by geodesics and metric ball boundaries. We also formulate a characterization in terms of the so-called L'evy net produced by a metric exploration from one measure-typical point to another. This characterization is part of a program for proving the equivalence of the Brownian map and the Liouville quantum gravity sphere with parameter γ=8/3.

Description

Keywords

Brownian map, Brownian snake, Brownian tree, Brownian disk, random planar map, Liouville quantum gravity

Journal Title

Journal de l'Ecole Polytechnique - Mathematiques

Conference Name

Journal ISSN

2429-7100
2270-518X

Volume Title

Publisher

Cellule MathDoc/CEDRAM

Rights

All rights reserved